Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1799 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,18,19,34)(5,22)(6,25,27,11)(7,12)(8,16)(9,21)(10,24)(14,38,41,61)(15,36)(17,45,43,37)(20,50,49,48)(23,33,55,53)(26,31)(28,57)(29,42)(30,44,56,63)(32,54)(35,40)(39,47,62,59)(46,60)(51,52)(58,64);(1,3,5,16,7,15,24,42)(2,8,10,13,12,29,22,36)(4,14,20,39,19,41,49,62)(6,17,23,44,27,43,55,63)(9,28,32,35,31,58,52,60)(11,30,33,37,25,56,53,45)(18,47,48,38,34,59,50,61)(21,40,51,57,26,46,54,64);(1,4)(2,9)(3,14)(5,20)(6,26)(7,19)(8,28)(10,32)(11,34)(12,31)(13,35)(15,41)(16,39)(17,46)(18,25)(21,27)(22,52)(23,54)(24,49)(29,58)(30,59)(33,50)(36,60)(37,61)(38,45)(40,43)(42,62)(44,64)(47,56)(48,53)(51,55)(57,63);(3,15)(4,19)(8,29)(9,31)(13,36)(16,42)(17,43)(18,34)(20,49)(21,26)(30,56)(32,52)(37,45)(44,63)(48,50)(51,54)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303777, #72303803, #72303806, … |
| maximal normal subgroup of | #72303777, #72303803, #72303806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400934);